Problem 107 - A Graphical Approach to College Algebra

Question

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=\frac{2}{3} y^{2}-4 y+8$$

Step-by-Step Solution

Answer

Vertex: (2, 3); Axis: y = 3; Domain: [2, ∞); Range: (-∞, ∞).

Step 1: Identify the Form of the Equation

The given equation is \(x = \frac{2}{3}y^2 - 4y + 8\). This represents a parabola that opens horizontally because the variable \(y\) is squared, not \(x\). This indicates a parabola opening to the right or left because the equation is not in the standard \(y = ax^2 + bx + c\) form.

Step 2: Convert to Standard Form

To find the vertex and axis, we convert the equation into the standard form for a horizontal parabola: \(x = a(y - k)^2 + h\). First, complete the square on the \(y\) terms: \(x = \frac{2}{3}(y^2 - 6y) + 8\). Take half of \(-6\), square it, and add and subtract inside the parentheses:\(y^2 - 6y + 9 - 9 = (y-3)^2 - 9\). The equation becomes: \(x = \frac{2}{3}((y-3)^2 - 9) + 8\).

Step 3: Simplify the Equation

Use distribution to simplify:\[x = \frac{2}{3}(y-3)^2 - 6 + 8\].Simplify further to get:\[x = \frac{2}{3}(y-3)^2 + 2\]. This is now in the standard form \(x = a(y-k)^2 + h\), indicating the vertex \((h, k)\).

Step 4: Determine Vertex and Axis

From the standard form \(x = \frac{2}{3}(y-3)^2 + 2\), the vertex is \((2, 3)\). The axis of the parabola is the horizontal line \(y = 3\), as the parabola opens horizontally.

Step 5: Determine Domain and Range

The parabola opens to the right because \(\frac{2}{3} > 0\). Thus, the domain is \([2, \infty)\), as \(x\) values start at 2 and extend to positive infinity. The range is all real numbers \((-\infty, \infty)\) because \(y\) can take any value.

Step 6: Graph the Parabola by Hand

Plot the vertex \((2, 3)\) on the coordinate plane. Since the parabola opens to the right, sketch the curve from the vertex moving rightward symmetrically along the axis \(y=3\). Check key values like \((2, 4)\) and others to ensure smooth curvature.

Step 7: Verify Using a Graphing Calculator

Input the equation \(x = \frac{2}{3}y^2 - 4y + 8\) into the graphing calculator and observe the graph's accuracy. Ensure that the vertex and direction match your hand-drawn curve.

Key Concepts

Vertex

The vertex of a parabola is the point where it bends. In the case of our horizontal parabola, the vertex represents both the minimum or maximum value depending on its orientation.

To find the vertex, transform the equation into the standard form of a horizontal parabola: \(x = a(y-k)^2 + h\). After completing the square, our equation is \(x = \frac{2}{3}(y-3)^2 + 2\).

From here, we easily identify the vertex as \((h, k) = (2, 3)\). This means our parabola turns around at the point \( (2, 3) \) on the graph. It's where the curve shifts direction.

• The vertex acts as a reference point when sketching the curve.
• For horizontal parabolas, it's crucial in determining the range and axis of symmetry.

Domain

The domain of a function tells us all the possible input values (in this case, "x" values) that make the equation work. For a horizontal parabola, this depends on its direction of opening.

Since our parabola opens to the right (indicated by the positive coefficient \(\frac{2}{3}\)), the smallest x-value begins at the vertex, i.e., 2, and extends infinitely in the positive direction. Thus, the domain is

• \([2, \infty)\)

Understanding the domain helps in predicting the spread of the parabola along the horizontal axis.

Range

The range of a function describes all possible output values, which pertain to the "y" values on the graph. For horizontal parabolas, this range is more straightforward.

In our horizontal parabola, the range is all real numbers because there's no bounded limit on the vertical values. The parabolic curve can stretch indefinitely up (and down) along the y-axis. Thus, the range here is:

• \((-\infty, \infty)\)

The range highlights the vertical reach of the curve and reaffirms that for horizontal parabolas, y-values are unlimited.

Horizontal Parabola

A horizontal parabola is defined by having its main variable squared in terms of another, i.e., \(x = a(y-k)^2 + h\). Unlike the vertical parabola that opens upwards or downwards, a horizontal parabola opens to the right or left. This opening depends on the sign of the coefficient in front of the squared term.

Our equation, \(x = \frac{2}{3}y^2 - 4y + 8\), becomes \(x = \frac{2}{3}(y-3)^2 + 2\) when in standard form, reflecting a horizontal orientation.

• If \(a > 0\), it opens to the right.
• If \(a < 0\), it opens to the left.

For our specific equation, \(\frac{2}{3} > 0\), demonstrating that the parabola opens to the right. Horizontal orientation impacts how we describe its domain and range, and sets it apart from the typical vertical parabola many students are familiar with.